Shape Derivatives for the Compressible Navier-Stokes Equations in Variational Form

TL;DR

This paper investigates shape derivatives for the compressible Navier-Stokes equations, comparing approaches based on surface gradients and variational forms, and demonstrates the superior accuracy of the variational method in a DG solver.

Contribution

It introduces and compares two methodologies for shape derivatives in compressible flow optimization, highlighting the advantages of the variational approach.

Findings

Variational shape derivatives yield more accurate gradients than finite differences.

The gradient expressions from different formulations are equivalent only under strong form solutions.

Implementation within a Discontinuous Galerkin solver confirms the effectiveness of the variational approach.

Abstract

Shape optimization based on surface gradients and the Hadarmard-form is considered for a compressible viscous fluid. Special attention is given to the difference between the 'function composition' approach involving local shape derivatives and an alternate methodology based on the weak form of the state equation. The resulting gradient expressions are found to be equal only if the existence of a strong form solution is assumed. Surface shape derivatives based on both formulations are implemented within a Discontinuous Galerkin flow solver of variable order. The gradient expression stemming from the variational approach is found to give superior accuracy when compared to finite differences.